Maria was 500 kilometres east of the northeastern coastal town of Yilan with gusts of up to 190 kilometres an hour as of 10 am local time (0200 GMT), the weather bureau said.
Its impact was expected to be the strongest from late Tuesday to early Wednesday, with downpours up to 500 millimetres (19.7 inches) forecast in some areas, the bureau added.
Officials have warned of possible floods and mudslides.
Taiwanese authorities announced that offices and schools would close on
Tuesday afternoon in five cities, including Yilan, where strong waves have started to pound the shore.
Local television footage showed farmers in Yilan rushing to their fields harvest scallions, the county's most famous produce.
Fishermen in coastal Keeling also brought in their catch and secured their boats ahead of the storm, one day after dozens of ferry services to outlying islands were cancelled.
He urged people to stay indoors and cooperate with the government's disaster prevention plans.
The typhoon will not make a direct hit if it continues on its current trajectory, which would see it skim off northern Taiwan, according to the weather bureau.
Taiwan is frequently hit by typhoons in the summer. Last year more than 100 people were injured when Typhoon Nesat battered the island, causing flooding and widespread power outages.
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